Green's third straight 3-pointer made it 45-28 about 5 minutes into the second quarter, and it seemed the trend of blowouts would continue.
But James suddenly got rolling during a 14-2 Miami spurt that cut it to five on his third consecutive Heat basket.
San Antonio made 21 of 34 shots (62 percent) in the first half, opening a 61-52 lead on Parker's drive with 0.4 seconds left.
Miami then ran off eight in a row to start the second half and get within one. They cut it to one again later in the period before Ginobili led the flurry that finished the Heat for good.
It was a fitting finish if it was the last home game in the finals for San Antonio's star trio, which has combined for 101 playoff victories. Ginobili has said he might think about retirement as he turns 36 next month, and Duncan is 37.
Both coaches said it was difficult waiting two days between games — Popovich said it was "like death" — though he did say it was great for the Spurs because they have some older players.
The break seemed to help his team early, particularly Parker, whose energy sagged in the second half of Game 4 as he struggled with a strained hamstring that he said could tear at any time and would've had him sidelined during the regular season.
If things fall right for the Spurs, he'll have plenty of time to heal after Tuesday.
Notes: The last team to lose Game 5 of a 2-2 series and then win the title was the Los Angeles Lakers, when they beat Boston in 2010. ... The Spurs said Sunday that reserve guard Patty Mills had surgery to remove an abscess Friday and would miss the rest of the series. Mills had an infection in his right foot and the abscess developed between his fourth and fifth toes.